Modeling of Electrochemical Cells: HYD Lecture 10. Atomistic and Coarse-Grained Models of Polymers

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1 Modeling of Electrochemical Cells: Proton Exchange Membrane Fuel Cells HYD Lecture 10. Atomistic and Coarse-Grained Models of Polymers Dept. of Chemical & iomolecular Engineering Yonsei University Spring, 2011 Prof. David Keffer Prof. David Keffer

2 Lecture Outline Atomistic simulations method sample results (Wang, Q., Keffer, D.J., Petrovan, S., Thomas, J.., Molecular Dynamics Simulation of Polyethylene Terephthalate Oligomers, J. Phys. Chem.. 114(2) 2010 pp ) Coarse-graining Procedure (Wang, Q., Keffer, D.J., Nicholson, D.M., Thomas, J.., Use of the Ornstein Zernike Percus Yevick Equation to Extract Interaction Potentials from Pair Correlation Functions, Phys. Rev. E 81(6) 2010 article # ) Coarse-grained simulations method sample results (Wang, Q., Keffer, D.J., Nicholson, D.M., Thomas, J.., Coarse grained Molecular Dynamics Simulation of Polyethylene Terephthalate (PET), Macromolecules 43(24) 2010 pp )

3 Motivation The simulations presented thus far in the course have focused on the structure and dynamics within the aqueous phase. The movement of water and ions occurs on a much faster timescale than the movement of the polymer. Thus, in those analyses, the polymer is considered to be essentially rigid. If we are interested in understanding processes that require polymer mobility, such as the response of the membrane to a sustained applied electric field, then we must implement simulations that are capable of much longer timescales. This is one motivation for moving to coarse-grained simulations of polymers.

4 Molecular l Level lsimulation i : Atomistic MD simulation: Small system, short time. number of oligomers 125 to 216 number of atoms 4320 to simulation duration 1 to 100 ns system size ~ 6 nm The use ofperiodic boundary conditions allows bulk properties to be extracted from these simulations.

5 Can t Simulate long chains hi with ihatomistic i Model Why? Longest relaxational time of polymers scales with chain length to the 3 4 power. PET Relaxation Times DP time (ns) For long chains (DP > 10), the relaxation time is longer than the maximum simulation time available on modern computational resources What is the solution? Coarse grained models

6 Molecular l vs. Coarse Grained dmodels: Example Polymer: PET Molecular Model Coarse Grained Model A atomistic detail particles are molecular l groups structure is known grouping is based on intuition interaction potentials known interaction potentials unknown many degrees of freedom computationally expensive limited to short chains fewer degrees of freedom computationally modest works for long chains use for DP 1 to 10 use for DP 25 to 50

7 Procedure 1. Run molecular level simulation for DP = 1 to Parameterize coarse grained potential based on molecularlevel simulations 3. Run coarse grained simulations for DP = 25 to 50

8 Molecular l Dynamics Simulation: i ased on solving Newton s equation of motion To simulate N atoms in 3 D, you must solve a set of 3N coupled nonlinear ordinary differential equations. F ma The force is completely determined by an interaction potential. F U The ODE for particle i in dimension is thus d x i, 1 U dt m x 2 i, We must provide an interaction potential from either theory, quantum mechanical calculations or experiment.

9 Molecular l Dynamics Simulation: i Intramolecular l potential il We use the anisotropic united atom (AUA) potential model developed by Hedenqvist, haradwaj and oyd for PET to describe the intramolecular and intermolecular potential of HET molecules. Namely, all atoms of HET molecule are explicitly represented expect for the hydrogen connected to carbons θ 5 4 1: bond stretching 2: bond bending 3: bond torsion 4: out of plane bending 5: intramolecular L J and electrostatic potential (for atoms over four bonds) Note: not all pairs are listed O: red; C: grey; H: white Hedenqvist et al. Macromolecules (neglected OH interactions) OH interactions from in Chen et al. J. Phys. Chem., 2001

10 Collaboration with Oak Ridge National Laboratory To solve the thousands of ODEs (largest system thus far is 55,000), we use the massively parallel supercomputers at ORNL, both Cray and IM. National Center for Computational Science Today the computing resources of the NCCS are among the fastest in the world, able to perform more than 119 trillion calculations per second. These resources are available to researchers at UT through discretionary accounts of the program directors.

11 Molecular Dynamic Simulation Snapshots all molecules shown selected molecules shown Snapshots of PET hexamer at T = 563 K and p = 0.13 kpa.

12 Molecular Dynamic Simulation Snapshots dimer trimer tetramer hexamer octamer decamer All snaphots at T = 563 K and p = 0.13 kpa.

13 Structural properties: Relaxation Times monomer Two traditional models: both fit well <R(t).R(0)>/<R R 2 > MD simulation fit to KWW model fit to Rouse model 0.4 exp(-1/τ R ) β exp(-(t/τ KWW )) time (ns) Tsolou, V.G. et al. Macromolecules, 2005 MD simulations provide characteristic ti relaxations times through h the analysis of the decay of structural auto correlation functions. For polymers, longest relaxation time is due to rotation.

14 Structural properties: Relaxation Times DP time (ns) end to end dista ance auto correla ation function monomer dimer trimer tetramer hexamer octamer time (ns) Characteristic relaxations times increase dramatically with DP, creating difficulties for MD simulation of longer chain molecules.

15 Structural properties: End to end Pair Correlation Function end to end pair corre elation function function end to end pair correlation f monomer dimer trimer DP <R > (Å) ete tetramer hexamer octamer trimer tetramer hexamer octamer distance (Angstroms) ± ± ± ± ± ± distance (Angstroms) The long tail of end to end pair correlation function is typical for polymer molecules, indicating the complexity of chain molecules.

16 Structural properties: Hydrogen onding Hydrogen onding as a function of DP monomer dimer trimer tetramer hexamer octamer g OH (r) distance (Angstrom) While the number of alcohol end groups decreases with DP While the number of alcohol end groups decreases with DP, the probability of finding the end groups forming hydrogen bonds increase with DP.

17 Thermodynamic properties: Density Melt density as a function of DP DP < ρ> > (g/cm 3 ) ±0.006 density (g/cm^3 3) monomer PET melt ± ± ± Need experimental references. Is bulk the right word? Report different state points ± ± ±0.034 DP The density does not change much as DPincreases, the melt density of PET is around 1.2 (g/cm 3 ) (ASTM D1238,T=285 ), our simulation results are comparable to the experimental data.

18 Thermodynamic properties: Enthalpy Enthalpy per molecule as a function of DP H (aj/molecu ule) enthalpy per molecule enthalpy per DP DP <H> DP (kj /mole) ± ± ± ± ± ± ± The enthalpy per molecule increases with DP. The enthalpy per repeat unit also increases with DP.

19 Thermodynamic properties: Heatcapacity (C p ), isothermal compressibility thermal expansion coefficient Fluctuations method Finite difference method: (one simulation): (multiple simulations) 1 1 V p+ε,t -V p-ε,t 2 V Vk T 1 (V)(H+PV) 2 Vk T C P (H+PV) Vk T β= - V(p,T) 2ε 1 V p,t+ε -V p,t-ε α= V(p,T) 2ε C= p H p,t+ε -H p,t-ε 2ε Give a small perturbation of density and temperature. Calculate changes of pressure, volume, and enthalpy. Derivative is the corresponding thermodynamic property. The fluctuation method for these thermodynamic properties is based on the variance of volume, enthalpy and covariance of volume and enthalpy of the system, which can be obtained from one MD simulation.

20 Thermodynamic properties: Heatcapacity (C p ), isothermal compressibility(β), thermal expansion coefficient (α) Finite difference : very close to the experimental data Fluctuation: ti too much uncertainty. Note: experimental ldt data is tk taken from Polymer Handbook written by J. randrup, E.H. Immergut, and E.A. Grulke 1999.

21 Transport properties: Diffusivity Diffusion coefficient is calculated from the center of mass mean square displacement DP D (m 2 /sec))x10 11 log (<(R (t)-r cm (0))2 >( 2 cm )) nomomer(slope:1.00) dimer (slope: 1.00) trimer (slope:1.01) tetramer (slope:1.00) hexamer (slope:1.01) octamer (slope: 1.01) ± ± ± ± 4.01 ± ± ± ±0.60 log (simulation time (fs)) The slope of the curve equal to 1 means our simulations are long enough to guarantee a good diffusivity. The self diffusion coefficient decreases with DP as the diffusion process becomes harder for longer molecules.

22 Transport properties: Viscosity Zero shear rate viscosity is obtained by integrating momentum autocorrelation function over time, by following Green Kubo relations. 5e-9 5e-9 n function tum auto correlatio moment 4e-9 3e-9 2e-9 1e-9 momentum auto correlation functio on 4e-9 3e-9 2e-9 1e-9 0-1e-9-2e-9 time (ps) monomer dimer trimer tetramer hexamer octamer -3e The decay to zero ensures convergence of the integration. 0-1e time (ps)

23 Transport properties:viscosity Zero shear viscosity as a function of DP. viscosity (pa.s) monomer dimer trimer tetramer hexamer octamer tetramer trimer octamer hexamer DP η(pa.s) x ± ± 5.45 ± ± ± 6.51 ± monomer dimer time (ps) ± ± ±8.0 The viscosity increases with DP, this is true based on polymer physics. We measured the viscosity of the monomer using a rheometer at UT. The experimental viscosity of the monomer at T=533 K and 1 atm is 3.8 x 10 3 Pa.s. Compare to a simulated viscosity at T=563 K and 1 atm of 3.83 x10 3 Pa.s.

24 Transport properties: Stokes Einstein equation Relationship between diffusion coefficient and viscosity R kt 4 D D kt 1 4 R ln (Rg) & ln (R) radius of gyration Rg(MD) R(MD) fit (R(MD)) fit (Rg(MD)) Once we know R, either D or η as a function of DP, we can calculate the other by using above equation ln (DP) For monomer and dimer, R agrees with R g, R increases with DP, fluctuating around its trend line with slope equals to 1. This is governed by scaling behavior of D and η.

25 Transport properties: Thermal conductivity ased on integration of heat flux auto correlation function over time. 2e-4 heat flux au uto correlation fu unction 2e-4 1e-4 5e-55 0 monomer dimer trimer tetramer hexemer octamer elation function heat flux auto corr 2.0e-4 1.5e-4 1.0e-4 5.0e e-5-1.0e-4-1.5e time (ps) -5e time (ps) The heat flux auto correlation function has a faster decay compared with the momentum auto correlation function.

26 Transport Properties: Thermal Conductivity thermal con nductivity (W/m m.k*10^-7) 4e-8 3e-8 2e-8 1e-8 monomer dimer trimer tetramer hexamer octamer octamer trimer monomer tetramer hexamer dimer DP λ(w/m.k) ± ± ± ± ± ± time ( ps) ±0.01 The thermal conductivity obtained from simulation is between 0.1 and 0.3(W/m.K), consistent with the experimental data (0.24 from polymer handbook).

27 Scaling Analysis Objectives: The structural, thermodynamic and transport properties all scale with DP according to polymer dynamic theory, these scaling behaviors enable us to verify our simulation results and give predictions to the properties of longer chains, such as DP=10,which isdifficultto to obtain frommdsimulation simulation. Principles: Study how the physical quantities change when DP is changed. Properties such as R g, R ete, kww, C p, D are all a(dp) b, The scaling exponent b is the constant we are interested in and different from property to property D a DP b Method: y fitting our simulation data to a(dp) b in log log plots, the slope of the curve is b. The literature reported values of b for polymer melts are as follows: b = 2.1 for D (reptation model) b = for R g, R ete b =1 for (below l critical molecular l weight) iht) M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, G. Tsolou, V.G. Mavrantzas, D.N. Theodourou, Macromolecules, 2005.

28 ScalingAnalysis: Chain radius of gyration (R g ) and average end to end distance (R ete ) 5 ) and ln (R g ) 4 3 R g (fit), slope=0.594 R ete (fit),slope=0.571 R g (MD) R ete (MD) Estimated value for decamer: <R g > =15.19 (Å) <R ete > =35.65 (Å) ln (R ete ) ln (DP) The scaling exponent for R g and R ete is and separately, very close to theoretical prediction of Our result is valid and we also give the prediction of the value of R g and R ete for decamer.

29 ScalingAnalysis: Heat Capacity (C p ) MD fit (slope=-0.081, R2=0.99) b=0.081 from MD ln ( Cp) 0.9 no reported literature value Estimated value for decamer: C p =2250 (J K 1 kg 1 ) ln (DP)

30 ScalingAnalysis: Diffusion Coefficient (D) MD fit (slope=-2.01, R 2 =0.99) MD simulation b = 2.01 ln (D) theoretical value: b = ln (DP) Estimated value for decamer: D =6.40x ( m 2 /sec)

31 ScalingAnalysis: Zero Shear Rate Viscosity (η) MD fit ( slope=0.96, R 2 =0.85) MD value b = 0.96 ln (viscosity) Literature reported -5 value is 1 (for PET with molecular weight -6 under critical point) ln (DP) Estimated value for decamer: η =2.24x10 2 (pa.s)

32 Coarse Grained model: based on atomisticmd simulation: Atomistic MD simulation is limited to DP up to 10, CG MD simulation will allow us to get to DP from 25 to 50 or even longer chains by neglecting gsome local degrees of freedom. A A A A A Atomistic model and com of CG beads CG model A CG model on top of atomistic model for hexamer (DP=6) CG chain for hexamer, total interaction centers is 13 for each molecule ( original:90).

33 Coarse Grained Potentials: parameterization We must figure out all interaction modes according to the CG model and parameterize the CG potentials for each mode (stretching, bending, torsion, nonbonded interactions) to match the structures of atomistic simulation. A AA bend A A A stretch CG CG CG CG CG U U U U U AA torsion A bend CG chain model and intramolecular interaction for hexamer, 1: bond stretching (A); 2: bond bending (A and AA); 3: bond torsion (AA) A A A stretch bend torsion non bond CG CG U (, r T) k Tln P (, r T) C stretch stretch r CG CG U (, T) k Tln P (, T) C bend bend CG CG Utorsion (, T ) k T ln Ptorsion (, T) C To get the potential of each mode, we have to obtain the distribution function first, this can be extracted by analyzing atomistic simulation configurations of short chains.

34 Coarse Grained potentials: distributions of CG beads We obtained P(r), P( ), P( ), g(r) based on atomistic MD simulation configurations. P(r) A stretching tetramer hexamer octamer decamer tetramer hexamer octamer decamer P( A-tetramer AA-tetramer A-hexamer AA-hexamer A-octamer AA-octamer A-decamer AA-decamer bending AA A separation r (Angostroms) (Angstroms) bending angle (degree AA tetramer hexamer octamer decamer non bonded P( ) torsion g(r) tetramer hexamer octamer decamer torsion angle degree separation (Angstroms)

35 Coarse Grained potentials: OZPY 1 method for CG polymers OZ integral equation: exact relationship between pair correlation function (PCF) and interaction potential g r 1 c ( r ) n c s h t d V A A A 1 A A 3 2 A OZPY: given U, find PCF OZPY 1 : given PCF, find U need to know: PCFs allowable combinations (A) n linear chain various combinations of stretching, bending and torsion interactions For this model, there are 34 allowable combinations that contribute to the indirect portion of the correlation.

36 Coarse Grained potentials: bonded and non bonded.

37 Coarse Grained MD simulation: comparison of structures. We compared the P(r), P( ) and P( ) from CG and atomistic MD simulation. istance distribution stretching di tion to orsion angle disbribu A A target CGMD separation (Angstroms) AA-target AA-CGMD AA torsion angle (degrees) istribution bending angle d A-target A-CGMD AA-target AA-CGMD AA bending angle (degrees) A The distributions of beads with bonded interactions from MD simulations match relatively well.

38 Coarse Grained MDsimulation: comparison ofstructures comparison of non bonded PCFs (, A and AA) from CGMD and atomistic MD simulations nonbonded distance disyribution target -CGMD nonbonded distance distribution AA AA-target AA-CGMD distance (Angstroms) distance (Angstroms) ribution nonb bonded distance disbr A distance (Angstroms) A-target A-CGMD The non bonded PCFs from CGMD and atomistic i MD simulations i match fil fairly well. Solving the integral equations provides us a way to get a reliable non bonded interaction potential for CGMD simulation.

39 Coarse Grained MD simulation: chain end to end distance distribution bution pr robability distri DP = 10 DP = DP = DP = 40 DP = Decamer: Comparison of chain end to end distance distributions from CGMD and atomistic MD match fairly well distance (Angstroms) The distributions are not typical Gaussian distributions, with chain length increases, they become more Gaussian like.

40 CGMD simulation: comparison of dynamic property chain end to end distance auto correlation function (ACF) Note: We must scale time in CGMD simulation (Harmandaris et al. Macromol. Chem. Phys. (2007)) (based on ratio of the values of selfdiffusivities) Atomistic MD: ACFs for long chain systmems can not reach 0 in short times For Atomistic MD: ACFs for long chain systmems can not reach 0 in short times. For decamer, it took roughly 6 months to finish a run of 30ns CGMD: simulations of the same systems. Apparent speed up is observed. All ACFs can reach 0 in short times. For decamer, it took just 2 weeks to finish a run of 180 ns.

41 Scalingfactor and Scalingexponent (b) Scaling factors: based on the ratio of the values of self diffusivities from atomistic and CG simulations. The value 5.38 is used to scale the dynamic properties back to atomistic scope. Scaling exponents: b from polymer physics. DP 1~10 X a Simulation method Atomistic MD DP b D KWW <R ete > <R g > ~10 CGMD ~50 CGMD Rouse Model (1, 2) Reptation Model (1, 2) N/A N/A Tzoumanekas et al. Macromolecules Lahmar et al. Macromolecules 2009

42 Coarse Grained MD simulation: Scaling exponents of R ete and R g 6 ln (R ete ) and ln (R g ) for Long chains: obtained b is close to 0.5 for short chains: obtained b is in the range of 0.57~ ln (DP) The values for both short chains hi and long chains hi agree with the Rouse and Reptation theory respectively.

43 Coarse Grained MD simulation: Scaling exponent of self diffusivity i it Rouse model: for short chains, b = 1 ln(d(m 2 /s)) Reptation model: for long chains, b = ln(dp) Obtained b deviatesfrom Rousetheoryfor shortchains, agrees with reptation theory for long chains.

44 Coarse Grained dmd simulation: entanglement tanalysis To further understand the reptation behavior, we can do entanglement analysisby extracting entanglement information directly from configurations of the chains. The Z code: A common algorithm to study the entanglements in polymeric systems (Kroger, M. Comput. Phys. Commun ) What can we get from Z code? mean contour length of primitive path (<Lpp>), tube diameter (d), number of monomers between entanglement points (N e ), interentanglement strand length (N ES ), defined as: N ES Z N ( ( N N 1) 1) N Kamio et al. Macromolecules 2007, 40, 710.

45 Coarse Grained MD simulation: entanglement analysis DP <L pp > (Å) d (Å) N e Z N ES rheology models N/A 35 a, 30.2a, b 24.2 b, N/A 14.9 d 25.0 c For DP = 10, unentangled system. for DP = 20 to 50, (d), (Ne) and (N ES ) are very close to the reported values for entangled PET melts. a Fetters et al InPhysical Properties ofpolymers Handbook; James E Mark 2007 a Fetters et al. In Physical Properties of Polymers Handbook; James E. Mark, 2007 b Fetters et al. Macromolecules 1994, 27, c Lorentz, G.; Tassin, J. F. Polymer 1994, 35, d Kamio et al. Macromolecules 2007, 40, 710.

46 Conclusions: Atomistic simulations of PET from DP = 1 to 10 yield a variety of structural, thermodynamic and transport properties that compare well with experimental data. CG model of PET was implemented in CGMD simulations of PET chains with DP up to 50. This allows simulation up to 1000 ns. The CG potential is parameterized to structural distribution functions obtained from atomistic simulations using OZPY 1. The CGMD simulation of PET chains satisfactorily reproduces the structural and dynamic properties from atomistic MD simulation of the same systems. For the long gchains, we find the escaling exponents epoetsof 0.51, 05,050a 0.50 and 2.00 for average chain end to end distance, radius of gyration and self diffusivity respectively. Th t l t l i h th t t b di t (d) (N ) d (N ) The entanglement analysis shows that tube diameter (d), (Ne) and (N ES ) of long chain systems are very close to the reported values for entangled PET melts.